An example of a Z-PBW algebra which is not a PBW algebra?

I have just rechecked the assertion in the Example at the bottom of page 97 and to the best of my understanding it is correct.

I can imagine one cause of a possible confusion, namely, the variables are not listed in the order of their increase in the sense of the ordering that makes them Z-PBW-generators. The order should be $x^{sigma+1,sigma}>y^{sigma+1,sigma}>z^{sigma+1,sigma}$, the order of monomials being the lexicographical order corresponding to this order of generators. It is presumed that the surviving monomials in the PBW-basis are those that cannot be expressed as linear combinations of any smaller ones (modulo the relations).

Specifically, the quadratic monomials that do not survive (do not belong to the set $S^{sigma+1,sigma-1}$) are $x^{sigma+1,sigma}y^{sigma,sigma-1}$ and $x^{sigma+1,sigma}z^{sigma,sigma-1}$. Given their form, it is immediately clear that the PBW condition is satisfied.

UPDATE. Well, another source of a possible confusion is a misprint in our formulas. It should be $b_{sigma+1}=-1/c_sigma$ and $c_{sigma+1}=-1/(ab_sigma)$ and not the other way around. To put it simply, the constants $b_sigma$ and $c_sigma$ are chosen in such a way that the terms $x^{sigma+1,sigma}x^{sigma,sigma-1}$ cancel out in each relation.

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